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Handbook > Consumer Theory > Choice and Preferences > Derived Properties of Preferences Printer Friendly

Derived Properties of Preferences

There are two additional relations, called indifference and strict preference that can be defined in terms of the preference relation is at least as good as. Both of these relations inherit the properties of is at least as good as. In order to prove the existence of a utility function that represents choice, transitivity of the strict preference relation is needed.

Indifference

Two consumption levels x and x' are indifferent if x is at least as good as x' and if x' is at least as good as x. This is typically written x is indifferent to x'.

Indifference is transitive because the relation is at least as good as that it is derived from is transitive.

Proof: If x is indifferent to x' and x' is indifferent to x'', then from the definition of indifference above, x is at least as good as x' and x' is at least as good as x'' so that x is at least as good as x'', and also x'' is at least as good as x' and x' is at least as good as x so that x'' is at least as good as x. Taken together, this shows that if x is indifferent to x' and x' is indifferent to x'' then x is indifferent to x''.

Strict preference

Consumption level x is strictly preferred to consumption level x' if x is at least as good as x' and not x' is at least as good as x. (The symbol not mean "not".) In words, x is at least as good as x', but x' is not at least as good as x. This leaves only the possibility that x is strictly preferred to x', which we write as x is at least as good as x'.

Strict preference inherits transitivity from is at least as good as.

Proof: Let x and x' be such that x is preferred to x' and x' is preferred to x''. Then the strict preference relation is preferred to is transitive if this implies that x is preferred to x''.

From the definition of strict preference, x is preferred to x' if and only if x is at least as good as x' and not(x' is at least as good as x). Similarly, x' is preferred to x'' if and only if x' is at least as good as x'' and not(x'' is preferred to x'). From x is at least as good as x' and x' is at least as good as x'' and transitivity of is at least as good as we get x is at least as good as x''. Suppose that x'' is at least as good as x. Since x is at least as good as x' this would imply, again by transitivity of is at least as good as, that x'' is at least as good as x'. But x' is at least as good as x'' which by definition implies that not(x'' is at least as good as x'). This contradicts the assumption that x'' is at least as good as x, so x is preferred to x'' and not (x'' is at least as good as x), which is equivalent to x is preferred to x''.


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